# What does Rosenlicht mean by a “point”? By $k(v_1,v_2)$?

This is cross-posted from Math.SE at the recommendation of a commenter.

I'm reading M. Rosenlicht's 1956 paper, "Some Basic Theorems on Algebraic Groups" [link], and having trouble with some of the language and notation. In particular, I have two questions:

1) Rosenlicht defines an algebraic group as a variety with a group structure on the points. He pointedly does not assume the ground field $$k$$ is algebraically closed. As he was writing prior to Grothendieck's development of scheme theory, I am not sure how to interpret "points". In this context, does he mean the $$k$$-points? The $$\overline k$$-points i.e. the geometric points? Or what?

2) With $$G$$ an algebraic group, and $$V$$ a variety with an action of $$G$$, all defined over a field $$k$$, Rosenlicht frequently writes expressions like "$$k(g,v)$$" with $$g\in G, v\in V$$; for example, on p. 403, we have

We say $$G$$ operates on $$V$$ (or that $$V$$ is a pre-transformation space for $$G$$) if for each component $$G_\alpha$$ of $$G$$ we are given a rational map $$g\times v\to g(v)$$ of $$G_\alpha\times V\rightarrow V$$ such that if $$k$$ is a field of definition for $$G$$, $$V$$, and each of these rational maps and if $$g_1\times g_2 \times v$$ is a generic point over $$k$$ of $$G_\alpha \times G_\beta \times V$$ ($$G_\alpha$$, $$G_\beta$$ being any components of $$V$$) then

$$(1)\:\:\:\: g_1(g_2(v)) = g_1g_2(v).$$ $$(2)\:\:\:\: k(g_1,g_1(v)) = k(g_1,v).$$

What's meant by $$k(g_1,v)$$ here? I want it to be a residue field but it has 2 arguments. Is it the composite of the two residue fields? (In which case, the answer to question 1 cannot be "$$k$$-points"?)

Aside: I am particularly interested in Theorem 2 on p. 407 ("Rosenlicht's theorem"). However, my goal is to be able to read Rosenlicht's paper, so I would prefer a gloss of his usages over a reference to a more modern statement and proof.

Addendum: The accepted answer contains a link to another question/answer which links to this very helpful piece by Raynaud in the Sept. 1999 Notices, discussing Weil's foundational approach (which turns out to be the context for Rosenlicht's paper). I include the link here for the sake of self-containedness of the question/answer pair.

Weil's foundations (which Rosenlicht uses) use a universal domain, which is an algebraically closed extension of $$k$$ of infinite transcendence degree. There is a discussion of universal domains here. In this language, "point" means a point with coordinates in a universal domain. The expression $$k(g,v)$$ means the subfield of the universal domain generated over $$k$$ by the coordinates of the points $$g$$ and $$v$$.